Fluctuations, Moderate and Large Deviations about Hydrodynamic Limits of Space-Time Hawkes Processes

We study Gaussian fluctuations, moderate and large deviations around the high-density hydrodynamic limit of critical and nearly critical space-time Hawkes processes initialized by a prescribed Poisson field of past events, with no subsequent immigration. Under a finite mean reproduction delay and centered displacements with finite spatial covariance, a positive cluster-transform estimate yields a distribution-valued central limit theorem (CLT) and a moderate deviation principle (MDP) throughout the moderate range. For finite nonzero mass of the initial profile, we obtain a full large deviation principle (LDP) for the space-time event measure in the vague topology; its good rate function is a quadratic branching-control energy with a prescribed initial trace and a finite-mass domain. The lower bound uses an explicit approximation by smooth reproduction controls, and the upper bound uses a microscopic exponential martingale and a global generator approximation. We distinguish exact-mean centering from PDE centering and compute deterministic bias corrections under second delay and fourth spatial moments. For a factorized heavy-tail power-law delay, we establish the CLT, MDP, and full measure-valued LDP for the space-time measure, with finite nonzero mass of the initial profile for the LDP. Its fractional control energy retains the source generated by the prescribed past. A source-preserving regularization proves approximation of every finite-action measure. Compact-kernel bounds for initial profiles of infinite mass and the degenerate case of a zero initial profile are stated separately.